The Power of Math! (featuring Yukari Yakumo)
by DezoPenguin
Summary: The compilation of the Gensokyo Chronicle is serious business (very much unlike this omake), so when Akyuu has some confusion over some very un-youkai-like elements associated with a very youkai-like youkai, she decides to go straight to the source for answers. Which she sort of receives. (Now featuring additional omake featuring Yukari's continuing mathematical encounters!)
1. Chapter 1

Hieda no Akyuu hummed softly as she set out the tea things on the low table. The ninth Child of Miare wondered if she ought to wash the ink stains from her fingertips before her guest arrived, but decided that it probably wouldn't matter to Kozusu. The bookshop owner would probably appreciate the point, come to think of it.

"Hard at work again, are you?"

Akyuu yelped, dropping the teapot. It plunged towards the bare wood floor, only to plunge through a hole that suddenly appeared in midair below it. An eye seemed to wink at her from out of the gap before it closed again.

"I apologize for startling you."

Akyuu turned to the source of the familiar voice, which she'd already placed from memory as belonging to Yukari Yakumo. Sure enough, the tall, blonde youkai was already seated on the other side of the tea table. Last time she'd visited, she'd been wearing a violet Western-style dress with long white gloves; today, she had on a white one with deep purple insets and carried an ornate fan. With her other hand she extended the fallen teapot to Akyuu.

"I suppose that I should have announced myself properly at the front door, but the last time I paid a call that poor maid of yours nearly had a fit of nerves."

"Well, isn't that the appropriate response when face-to-face with a youkai?" Akyuu poured tea into one of the cups.

Yukari chuckled softly and accepted the offered tea. "Perhaps so. But you don't seem to be subject to such reactions."

"Well...I did want to see you, as a matter of fact, so I'm rather glad you called." She wondered for a moment how, exactly, Yukari _had_ known that Akyuu wanted to see her, given that she'd only mentioned it to Kozusu and not even to the Hakurei shrine maiden, but gave it up as a fruitless thought.

 _To say nothing of how she dropped in just in time for tea._

"Indeed?"

"It's about the _Chronicle_..." Akyuu began, then glanced down at her fingertips.

"A brush isn't quite as efficient as modern methods of printing, but don't you find it has an elegance all of its own?"

"Oh, yes! I didn't know that you were a writer, though, Miss Yukari."

The youkai sage offered a shrug.

"I dabble. Or did you mean, 'she rarely does things herself, leaving practically all of her work to a beast she controls'?"

Akyuu probably would have blushed at the unflattering reference to Yukari's laziness from the _Chronicle_ , had it not been for the fact that it was one of the edits Yukari had inserted to the volume.

"...Either one?"

Yukari smiled at her and took a sip of tea.

"In all seriousness, though, what would you like to know?"

"Well, it's a bit hard to ask, but it's about something that I wrote about you that made me curious. It's about mathematics."

"Oh?"

"I understand that shikigami would have a high degree of mathematical knowledge, but..." She trailed off again, finding it hard to say things like, _but what does a gap youkai use it for?_ It was easy enough to understand _that_ Yukari was considerably more intelligent than a human, but she didn't quite follow why it was in that particular area of study.

"It seems like an unusual hobby for a youkai?" Yukari suggested after letting her dangle for a moment.

"It rather does, if you don't mind?"

"No."

Akyuu blinked.

"Wait, did you mean 'no, it's not an unusual hobby,' or 'no, I don't mind you asking'?"

"Yes," Yukari said, and drank more tea. "It's good for that as well."

"What is?" Akyuu said, completely lost by now.

"But the main point is, without an advanced knowledge of mathematics and the ability to make high-speed calculations, I wouldn't be able to properly exercise control over boundaries," Yukari took pity on her.

"Really? It seems so unusual. After all, youkai are creatures that are inherently fantastical. Even Gensokyo itself exists because the boundary between reality and fantasy is embodied in the Great Hakurei Barrier, doesn't it? But math is, well, it's so grounded in specifics and rigorous rules, that it...doesn't seem very youkai-like."

Yukari chuckled softly.

"You're greatly underestimating mathematics if you think that, Miss Hieda. On the contrary, math is just as capable of creating entire constructs of rules and systems that exist to describe what does not and, indeed, _cannot_ exist, simply because it happens to be useful to do so. After all, if you think it is complex to make calculations concerning what exists, then how much more complex does it have to be when the world outside is telling you that the subject of your thoughts does _not_ exist?"

Akyuu wasn't entirely sure if she understood that, but it reminded her of something Ran Yakumo had once said, that her mistress would be capable of calculating the depth of the Sanzu River, even though the Sanzu River was bottomless.

"Consider even something as simple as one of my gaps," Yukari elaborated. "Imagine for a moment the calculations required of how, exactly, two discrete points in time and space are connected to one another, and where the boundary lies between _here_ and _there_ , to do something like reaching out and fetching fresh tea from your kitchen to replace what I'm drinking here." The air in front of her shimmered, and opened into another "hole" in space, into which she leaned, her head and shoulders vanishing.

~X X X~

Reimu Hakurei sighed and leaned back into the steaming water. Youkai, she decided, should be courteous enough to harass people by night. Four hours of cleaning up the byproducts of two rival teams of kappa engineers fighting over the best way of finishing Kanako's latest scheme to bring science to the masses (and why wasn't Sanae exterminating problems in her own backyard, anyway?) left a shrine maiden tired to the bone, and it wasn't even seven at night yet.

Still, the problem was dealt with, and Reimu now had a hot bath, to say nothing of a full bottle of equally hot sake to warm her insides to match. There was a lot to be said for the simple comforts of home.

She wondered what it said about her life that she'd become so familiar with the sensation of the fabric of reality being ripped apart in her immediate proximity that she could recognize it immediately when it happened.

" _Dammit, Yukari!_ "

~X X X~

The youkai sage's upper body jerked backwards out of the gap and she fell over, the wooden washtub that had apparently been the cause of her sudden repulsion (and her bleeding nose) rattling off the floor before the gap snapped shut.

"Miss Yukari, are you all right? What happened?" Akyuu gasped.

"...Forgot to carry the two."

~X X X~

 _A/N: Yukari's line, "It's good for that as well," is a reference to the "Mathematician's Answer": to answer either/or questions with yes/no answers (so-named because in math, "is it A or B?" would be answered "yes" if the answer was either A or B and "no" it was none of the above), which is what she was giving Akyuu._


	2. Chapter 2

**Omake Week 2015, Day 6:** _Yukari is fun as heck to write about, but I'm still surprised that I was able to come up with more than one example of math-based humor..._

~X X X~

Yukari Yakumo stepped out onto the veranda and gave a yawn and a stretch, feeling the joints pop in her arms and shoulders. Sometimes, she thought to herself, a youkai just felt her age. Though, she noted, she thought she was doing rather well given that that age was several millennia.

She had overslept this evening, something made possible because she'd given her shikigami Ran a long list of chores to accomplish that would take her all over Gensokyo. By Yukari's estimation, Ran would just then be about halfway up Youkai Mountain dealing with a tengu-kappa boundary dispute, rather than in Yukari's bedroom pretending to be an alarm clock.

It was, she had to admit, a lovely evening. No clouds obscured the star-strewn sky, and the moon was a crimson sliver against the chartreuse background. Idly, she considered slightly tuning the barrier that kept the Yakumo residence isolated from random wanderers to adjust the colors, then decided that the effort would probably not be worth it. And she was expecting Eirin Yagokoro to drop by in the early morning, and she was interested in how the transplanted Lunarian would find her way past the current setting.

Thus, Yukari decided, she would simply relax on the veranda with a cup of tea to start her evening. Of course, one problem with Ran being absent was that tea was not going to magically materialize. Adjusting the boundary between noise and silence, she projected her voice so that it echoed throughout the entire residence and grounds.

"Chen, could you please come back to the veranda?"

She knew that Ran's shikigami would be present; the tasks she'd set Ran were ones where the kitsune wouldn't bring Chen along with her, and Ran being Ran she would have assigned Chen certain tasks of her own to keep the nekomata from getting into too much trouble when unsupervised.

And yet, Chen did not appear.

"Chen?" Yukari raised her voice slightly, not actually all the way to "miffed" but somewhere past "curious." The patter of tabi-clad feet on the floorboards followed, and the youkai cat burst out through the doors.

"I'm sorry, Lady Yukari! I didn't hear you at first!"

"Ran is keeping you busy in her absence?"

"Uh-huh." Chen pouted. "She has me doing _math problems_."

Yukari blinked.

"Well, mathematical ability is necessary for a shikigami to properly fulfill her functions, so it's good that you're practicing."

"Yeah, but these are hard!"

"Perhaps I can be of help? Why don't you let me take a look at what you're doing…and you can prepare a pot of tea while I'm doing that. Perhaps the Darjeeling would suit."

Chen brightened at the offer.

"Yes, Lady Yukari!"

She scurried back into the house, then came right back with a piece of paper. Handing it over, she disappeared back into the building again. Curious, Yukari looked down at the page.

 _Five human bank robbers in the outside world flee the police into the forest near the Hakurei Shrine and stumble through the weak point in the Great Hakurei Barrier beneath the third cherry tree past the fourth maple that's been widening for months. Two of them really enjoy grilled lamprey, so they follow the smell to Mystia's stand and buy several skewers, for which they are safely guided to the Human Village. One robber follows the road successfully to the Hakurei Shrine and is returned to the outside world. The remaining two robbers blunder to the shore of Misty Lake. Remilia is feeling hungry and happens to attack and nearly drain one, who is later found unconscious by Reisen and brought back to Eientei for treatment. The final robber runs off into the woods in terror. Assuming a start time of 2:30 a.m. and that the robber is an adult male in typical health and the equivalent level of stress that a Gensokyo native would feel under the circumstances, calculate how dense the forest would have to be so that there would be a 75% chance Rumia would continue to crash into trees until dawn rather than be able to successfully catch and eat the last outsider._

Yukari smiled wryly, tweaked the boundary between knowledge and ignorance to print the calculations for the answer for Chen on the bottom of the page, and rose to her feet.

"Well played, Ran," she said, and flew away to shore up the weak spot in the barrier.


	3. Chapter 3

It was probably Sanae's fault. Or at least that's the way Suika would tell it when she sobered up sometime after lunch the next day.

The miko from Moriya Shrine had arrived to visit Reimu Hakurei to share plans for holding a festival among all of Gensokyo's shrines and temples, only to find Reimu missing and Suika sitting on the shrine steps, watching a single drop fall out of an upended gourd into her open mouth.

"Neverending sake gourd, my foot. I think this sake bug went on strike," she muttered. "Oh, hey there; I didn't see you come in."

"Good afternoon. Is Reimu around?"

"Nah, she had to fly up by your place, actually. Seems the tengu papers are having a bit of a circulation war. Something about a couple of newsies getting into a brawl right in the shopping street at the Human Village? Something like that, anyway. Anyway, she'll be stuck there until tomorrow, clearing it up."

Sanae winced. Technically, that was probably her job to fix that...though, on the other hand, she supposed that Reimu would probably be happy just to get the extra extermination business.

"So it's just you here, then?"

"Yeah, me and the shorty, there." Suika jerked a thumb in the direction of Shinmyomaru, who was napping stretched out on the veranda railing. She had to admit, in a way it was nice having the inchling living at the shrine, if nothing else than because it meant she wasn't the smallest one there any more.

Sanae grinned.

"So what are the two of you getting up to?"

"Up to?"

"Come on! When I was still living with my parents, any time they would spend the night away from home, that meant the chance to invite my friends over and party! I mean, what they don't know won't hurt them, right?"

Suika laughed.

"And here I thought you were supposed to be a good girl." She rubbed her hands together. "So who d'you think we can get to come?"

~X X X~

It had started out small. Just a couple of people for a quiet dinner and drinking party. But Gensokyo was a tiny community and news traveled fast even with the public press temporarily incapacitated. Suika was pretty well sure that they'd dragged every table from the shrine out into the courtyard and somehow three or four more had showed up anyway. Maybe they were tsukumogami who wanted to join the party or something.

In any case, the sake was flowing freely, and by two in the morning the party was showing no signs of slowing down even after the debate between Marisa and Nitori over whether magic or science made for the flashiest fireworks was cooled off while they were still fighting over which team Utsuho ought to be on. Sanae and Reisen were just coming back from putting away the water buckets when the Moriya Shrine miko demonstrated that yes, indeed, she was taking advantage of Gensokyo's lack of a drinking age by asking Eirin Yagokoro, "Hey, after all that, why don't you tell us a story? You must know a lot of good ones by now!"

Reisen sped to refill Eirin's cup, but found that someone else had gotten there first.

"Yes, why don't you tell us a story?" put in Yukari Yakumo. "I'm sure that you must know all manner of tales of ancient times."

Given that Yukari was quite possibly the oldest person at the party, the stunned silence that greeted this offhand provocation gave Eirin plenty of time to decide on an answer.

"Why yes, I think I will. As a matter of fact, Miss Yakumo, I'm reminded of a tale that calls back to the last time we drank together..."

~X X X~

The youkai had descended upon the Lunar Capital in waves. The unclean monsters of Earth, bearing with them the corruption of that tainted place and led by the youkai sage Yukari Yakumo, whose nefarious cunning had reached even to the hidden side of the moon by reputation, had posed a threat that even the astonishing powers and technology of the Lunarians would have had to work to defeat. Even the most haughty of the defenders believed that the struggle to throw back the invasion would be long and painful.

Or so it had seemed.

Perhaps it was the fundamental nature of these tainted creatures, the Lunarians considered. The initial waves were led by some of the most powerful and rapacious among the youkai, and yet their emotions had seemed to get the better of them, their desire to raven and consume drawing them out from their forces, making them rush ahead at the first contact with the enemy. Calm, rational military strategy began to fall apart quickly, and before the Lunarians' might, the youkai were cut down or driven back. By the time the battle should have been underway in earnest, it was over, and the youkai reserves were falling back without having ever having the opportunity to be committed to battle. Standing behind to cover the retreat of her forces, Yukari faced the leaders among the lunar emissaries, her dress torn and stained with the strife and smoke of combat.

"You have lost," Eirin stated flatly. The one-winged goddess by her side said nothing at all.

"It appears to be quite conclusive," Yukari said. A casual remark, as if she had been discussing the weather.

 _Conclusive, indeed,_ Eirin though. _You are better than this._ Which led, inevitably, to the conclusion that this invasion had never been about winning at all...or perhaps more accurately, that what Yukari meant by "winning" was something very different than what her fellow youkai did.

The emissary on the other side of Lady Sagume tossed her head.

"Of course! You filthy mud-dwellers could never breach the sanctity of our home!"

"Is that the lesson that should be carried away from this? How curious. Rather, it seems that we youkai cannot carry the day by force. But then, force, I think, would not be what the Lunar Capital should have to fear."

"What do you mean?" Sagume asked.

"As you have been so gracious as to offer a lesson, then permit me to give you one in return. You have turned us back today, but what do you think will happen when humans at last turn their eyes skyward?"

"They lack the magic to reach the moon in enough numbers to make a difference, even through the Spirit World," scoffed another of the escorts.

"True, but what of a crossing via physical space?"

Eirin's eyes widened. Of course, the Lunar Capital was isolated from the moon's surface, but if humans came to the moon regularly, saturating it with their spreading impurity, the life, the death, the _change_ that they brought with them everywhere they went, that barrier would not escape their gaze. And then, whatever happened would be an inevitable rush to disaster. Humanity's mere presence would be like an assassin's dagger aimed at the Lunarians' way of existence.

"Mind you, were the humans to be incapable of doing so until their imaginations became so decayed that they were closed to all but the physical world, it might be that they themselves would pass by in complete ignorance, hm? The signs of the direction they are taking are already there, and in only a few more centuries, I think the transfiguration may well be complete."

"But that's impossible!" Sagume blurted. "The calculations are far too easy! Why, it would only be a matter of a...a decade or two before they've determined how to step across space to the moon!"

"Really?" Yukari touched a fingertip to her lower lip, affecting a pose of curiosity. "I don't think that's going to be true any more."

~X X X~

Eirin raised the cup of sake to her lips and sipped placidly. Yukari's stock-in-trade was mystery and obscurity, as she herself had once found out to her detriment, and pulling back the veil on that mystery was a bit of good-natured revenge. And after all, if she was going to live as a human in Gensokyo, didn't that mean that she should be an enemy of youkai, in some symbolic way?

Of course, she had neglected to consider a few things. Most significantly, she had neglected to consider the fact that most of her audience was three sheets to the wind by now, and not really capable of comprehending the delicate subtleties of what Yukari had been doing.

She could almost see, for example, the little swirlies in Sanae's eyes as the living goddess tried to fit the mental pieces together.

"She..." she began, then turned towards Yukari. "You..."

"Yes?" Yukari said, a certain _flatness_ in her placidity that suggested that Eirin's shot had hit home...or that she was a couple of bottles further along than Eirin had calculated.

"You... _tricked_...Lady Sagume...into..."

"I would hardly say 'tricked,'" Yukari protested with mock innocence. "Indeed, I would say that she was quite content with the outcome."

Minor edits to character motivation were not, however, going to penetrate Sanae's understanding of the situation.

"You gap hag! _All the math homework I've ever done was your fault!_ " she screamed, and flung herself across the table at Yukari, scattering plates, cups, and bottles in her wake.

Eirin wasn't quite sure, she thought as soy sauce dripped off her nose and she cleaned the contents of a half-eaten plate of sushi off her bodice, but that the woman currently being throttled hadn't come out ahead after all.

~X X X~

Birds were greeting the early morning sun with merry chirping as Reimu walked over the threshold into the courtyard of the Hakurei Shrine. It showed every sign of being a pleasant day, and she was looking forward to relaxing at home after having ensured Gensokyo's freedom of the press without having to lose her temper.

Much.

And she was sure Aya and Hatate could get that one tengu unstuck from that tree-bole if they worked together. A teamwork-building exercise, and all that, to help celebrate their newfound peace. Why, taken in that light, it was probably a good thing, all told. That's right, she hadn't _really_ lost her temper; she'd...showed them the value of community and cooperation by giving them an outside issue to deal with! It was positively Yukarivellian of her, right?

 _Hell, Marisa might even buy it._

Her smirk, however, vanished in approximately twelve seconds, ten of which were spent just seeing the aftermath of the night's activities: sprawled bodies, dirty dishes, and empty bottles...the vast majority of which she clearly recognized as coming from the shrine's own cellar.

 _"Suika!"_

It said something about the quantity of alcohol consumed that most of the guests didn't even stir at Reimu's screech. The diminutive oni was not one of those, though, since she had the constitution of...well, an oni.

"Yeah? Oh, hey, Reimu," she said and yawned.

"This was your doing, wasn't it?"

"More or less, I s'pose. Hey, want a drink? I think I saved you some, around here...somewhere..."

 _"Do you have any idea how much all this sake cost?"_

Suika scratched her ear.

"Let's see...seventeen bottles of that kind...six of the crate you got from the kappa for fixing their dam, just the one out of Marisa's birthday present 'cause she wouldn't let anybody else have some...minus the two Reisen brought with her...plus six...an' figure the market value... Not really, no. But the way Eirin explained it, that's Yukari's fault."

"...Somehow, I'm not surprised."

~X X X~

 _A/N: Canon, at least at a superficial glance, seems to be a little fuzzy on the timeline of the first youkai invasion of the moon. But since_ Silent Sinner in Blue _says it was "hundreds of years" ago, that theoretically could fall into the period after Kaguya was exiled to Earth (c. 1000 y.a. in Heian-era Japan) but before Eirin went down to retrieve her when she was forgiven. And, honestly, for a gag fic, based nonetheless on an obvious-Sagume-joke-is-obvious like this one, "hey, it's possible" and "at a superficial glance" are pretty much going to be as far as I take it! (Also, yes, I know I fudged Suika and Shinmyomaru's relationship as being friendlier than canon for the sake of making a short joke.)_


	4. Chapter 4

It was, Daiyousei considered, a beautiful day on Misty Lake. The sun was shining, making the fog that gave the lake its name glitter like a wall of miniature rainbows, and the frogs were croaking on-key.

"Hey, Dai!"

Cirno's yell was a little _off_ -key, but then, she wasn't a frog. The other fairy came rushing up, her icicle wings fluttering as she hovered alongside her friend.

"You're pretty smart, right?"

"Umm..." Daiyousei was actually fairly intelligent for a fairy...intelligent enough to recognize the fact that _for a fairy_ was what humans called "damning with faint praise."

"So, I was just over near the Scarlet Devil Mansion and I heard the purple lady...you know, the one with the patches?"

"I think that's actually her name?"

"Anyway, she was talking to the sleeping dragon lady, and she asked her a math problem, and I don't know the answer."

Daiyousei blinked.

"Could you start from the beginning?"

Cirno sighed, but she was magnanimous to friends with lesser understanding, as was the righteous obligation of the strongest.

"The purple lady asked the dragon lady why she could never keep that black-white magician girl...I think it's the one I beat up all the time?...from stealing her books. And the green dragon lady got mad and asked why she kept blaming her when she was just the gate guard. So the purple lady said that trying to make excuses was as useful as asking how many angels could dance on the head of a pin. And I don't know!"

"Know what?"

"How many angels can dance on the head of a pin!"

"Is that what she was really asking?"

"I don't care about their silly argument, Dai. I just want to know what the answer to the math problem is!"

"Well, I definitely don't know."

Cirno rubbed her chin.

"That's good, I guess. At least there's no fairy better than I am at math! But still, what if one of the others finds out? Then I won't be the strongest, and that's not right, because I am, and—"

"You should ask someone," Daiyousei cut in before her head got too spinny trying to follow her friend's line of logic.

"But if nobody knows, what's the point in asking them?"

"Not a fairy. I mean a human or a youkai. It doesn't matter if _they_ know, right, because you'll still be the only fairy to know."

"Except you," Cirno said, narrowing her gaze. But then her expression cleared. "But that's okay, because you're just my sidekick on this quest! So who should we ask?"

"How about Alice? She's a magician, so she's pretty smart."

"Yeah! But the Forest of Magic is pretty far away..."

"She's right over there." Daiyousei pointed towards the shore. "She's probably going to visit Patchouli at the mansion."

"Perfect! Even destiny knows better than to get in my way!"

Cirno immediately zipped off towards the shore, Daiyousei tagging along. She wasn't really interested in the answer, but going on a quest sounded kind of fun, and Cirno's games were usually pretty entertaining when they didn't end up in her getting shot at.

Alice Margatroid was a youkai magician who specialized in dolls that she animated and controlled. She had a lot of different varieties, but the two that were hovering alongside her were fairly generic types, with red bows in their long blonde hair, blue dresses, white aprons, and red bows. They probably could charge with lances, shoot lasers, or just explode, so Daiyousei hoped that Cirno wasn't going to be too belligerent.

"Hey, doll lady!"

"Mmm? Oh, I remember you. You're the ice fairy who helped me test my Goliath Doll. Thanks to you, I got a lot of valuable data...though luckily we didn't need it since that giant youkai never appeared. Well, anyway, it was nice seeing you."

"Wait, I have a question for you!"

"Oh, you weren't just saying hello?"

"Yeah. You're smart, aren't you? I've got a math question I can't answer."

"Hmm...I suppose it wouldn't hurt. What is it?"

"How many angels can dance on the head of a pin?"

"How many...what?"

Alice was probably very much in tune with her dolls, Daiyousei thought, to be able to make an expression as blank and doll-like as that.

"How many angels can dance on the head of a pin!?" Cirno repeated, stamping her foot in the air. After all, it wasn't like she stuttered or had a weird accent that made her hard to understand.

"I don't think that's a math problem, Cirno."

"Huh? Why not?"

"Well, it sounds something more like a _koan_."

"That doesn't make any sense at all!" Cirno protested.

"What's a _koan_?" Daiyousei asked.

"Geez, Dai, even I know that. It's one of those pointy things that's round on the other end, like they put ice cream in. And some people call me an idiot," she muttered under her breath.

"Er, no," Alice said. "A _koan_ is a kind of riddle or statement that the Buddhists use to meditate on, to help lead them to enlightenment."

"But what does it _mean_?"

Alice shrugged.

"I don't know. I'm not a Buddhist. If you're really curious, maybe you should go ask at Myouren Temple."

Cirno smacked her fist into her palm.

"I'll do it! No dumb ol' Buddha is going to get away with keeping secrets from me!"

"...I don't think that's quite how that works, Cirno," Daiyousei said, but the ice fairy was already flying off.

The flight to Myouren Temple didn't take too long; apparently Cirno was interested enough in finding out the answer to her question that she didn't get distracted by any curious sights they happened across, and the fairies soon found themselves flying up to the gates of the temple. The place was bustling with activity, with numerous youkai monks flitting about on one errand or another. Daiyousei wasn't quite sure that this was what a Buddhist temple was supposed to be like, but then again most of the time when this many people were together in the Hakurei Shrine there was a drinking party going on, so she supposed maybe the problem was that she really didn't understand religion.

Byakuren Hijiri was in the main temple hall, reciting something from a scroll. The air here, unlike out in the corridors, was filled with a solemn stillness, as the only two other monks present were nearly silent as they went about their chores. The voice of the nun-turned-magician echoed off the rafters in soothing fashion as she recited the sutras.

"Hey! You! Monk lady!"

The sudden screech cut into Byakuren's concentration. She fumbled the scroll, desperately tried to grab it, but only caught one spindle with her left hand; the right-hand spindle bounced off her fingertips and went rolling away across the temple floor, unrolling a good thirty feet as it went. Byakuren turned, looked up, and smiled at Cirno.

"Good morning, Miss Fairy. How may I help you?"

Daiyousei's jaw dropped. _A saint! She must be a saint!_

"Alice told me you'd know the answer to a cone!"

"A _koan_ ," Daiyousei hissed.

"I just _said_ that, Dai."

"No, you...never mind."

"There are no proper answers to a true _koan_ ," Byakuren said. "They are tools by which we can focus our minds and come closer to the Buddha-nature."

"But it's a question. I mean, I mistook it for a math problem, so how can it not have an answer? That's just plain silly!"

A tiny furrow appeared between Byakuren's eyebrows.

"Why don't you tell me this _koan_ , and perhaps I can help to guide you to the enlightenment you seek?"

"Yeah! Enlightenment means answers, right? So...okay, how many angels can dance on the head of a pin?"

Byakuren blinked.

"How many...angels?"

"Yeah! Angels!"

She tapped her chin with one fingertip.

"I don't think that actually is a _koan_ , Cirno. It does sound like something similar, in that it appears to be an esoteric and philosophical question, so I can see why...Alice, did you say?...thought that it might be one, but in reality I think that it comes from another tradition entirely."

"Gah! You're making my head hurt!"

"She says we have to ask someone else," Daiyousei translated helpfully.

"It isn't a Buddhist saying; I can assure you of that. Perhaps it comes from one of the other religions? One of the miko might be able to help you."

"Okay. Thanks, monk lady!"

"Well," Byakuren murmured as the fairies flew away, "that was rather refreshing. Perhaps I'll have Shou ask the kappa if they can install air conditioning in the temple building?"

~X X X~

Reimu Hakurei was out in front of Hakurei Shrine when Cirno and Daiyousei arrived, sweeping dust and twigs off the veranda. It had been a particularly windy night, and dirt and debris had been scattered everywhere. Cirno's icicle wings stirred up a cloud as she buzzed in to a landing.

"Just what this shrine needs," Reimu sighed. "More fairies."

Daiyousei bowed politely. She still remembered the one time she'd had a spell card battle with Reimu.

"We're sorry to bother you, miss, but we need your help."

"Yeah!" Cirno spoke up. "Byakuren said that it wasn't a Bhuddist cone-thing, so we needed to go ask some other place!"

Reimu's gaze slid back over to Daiyousei.

"She means a _koan_. We're trying to figure out what it means."

"Yeah! And they're mean up at the Moriya Shrine, so we came here."

Reimu decided that she liked the idea that someone thought she was the nicer option than Sanae, even if it was a fairy (and even if it was really because of Suwako).

"All right, ask away."

"How many angels can dance on the head of a pin?"

"Huh?"

Cirno tapped her foot in the air impatiently.

"Like I said: how many angels can dance on the head of a pin?"

Reimu sighed. She then decided that wasn't quite good enough to express her feelings, so she facepalmed.

"That doesn't sound like a riddle; it sounds like a math problem."

"That's what _I_ said!" Cirno said proudly, sticking her chest out. "But Alice said it was a _koan_. But Byakuren said she'd never heard of it!

Reimu nodded. "Well, neither have I." She frowned, thinking. Having offered to help, she wanted to find a solution to the problem. The last thing she wanted was to find herself stumped by _Cirno's_ riddles, of all things. "All right, I think we ought to ask somebody who's good at math _and_ philosophy."

"Is it a long way away?"

Reimu stood up and stretched. "Well, I think we need to wait until after I have a bath."

"A _bath_?" Daiyousei boggled, but Reimu was already shooting her left hand up and back behind her head in a shoulder-twisting arc, vanishing into the tear in space that had suddenly appeared and came back with her hand fisted in the front of a purple dress, hauling a tall blonde in a mobcap out onto the _hisashi_.

"I'm getting too predictable," Yukari mumbled into the boards.

"Seriously, how do you do that?" Reimu sighed. "Ah, well, since you're here, make yourself useful."

Yukari sat up and clucked her tongue.

"So brusque today, Reimu. And by the way, it's actually a Christian saying."

"Wait, how long have you been listening?"

"And Alice was correct to liken it to a _koan_ , in that it is meant to be a philosophical riddle that leads one to enlightenment."

"But I wanna know the answer!" Cirno burst out.

"She has been wondering about it all day," Daiyousei pointed out. Given how long a fairy usually was able to keep a single thought in their head, that actually was fairly serious.

"Hmmm," Yukari pondered, then glanced at Reimu. "Well, since Miss Cirno is so serious about matters, it would be appropriate to find out."

"How are we supposed to...wait, are you suggesting...?"

Yukari's grin was positively evil.

"Precisely, my dear Reimu. The only way to properly answer that question...is to experiment." She cracked her knuckles.

"So which one of us fetches Tenshi, and which one of us gets the pins?"

~X X X~

 _A/N: I admit it, Yukari attempting to peep at Reimu is both obvious and getting old, but that segment also made my wife laugh herself silly, which by definition means it stays in! Also, those of you have been around since previous chapters will note that I finally surrendered and edited the name. It's supposed to be "The Power of Math! (feat. Yukari Yakumo)," but fanfiction-dot-net does not allow periods in its story titles (I think it may screw up their scripts for generating the URL for the story link, in all fairness...though the deletion of the second exclamation point is just them being annoying!), so "feat" without the period kept just bugging the heck out of me for its lack of grammatical precision because I am uptight about that stuff ("The Power of English!") so I gave in and changed the title to spell out the word, even if it screws up the allusion to music listings I was going for._


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